Shifting, Reflecting and Stretching Graphs

Definition Let $g(x) = f(u(x))$, where $u(x)$ is an expression in $x$. To find where the point $(a, f(a))$ of the graph of $f$ appears on the graph of $g$, solve $u(x) = a$ for $x$. Every solution $x_0$ gives the point $(x_0, f(a))$ on the graph of $g$, since $g(x_0) = f(a)$. Changes outside $f$, as in $c \cdot f(u(x)) + k$, then act on the height, sending $f(a)$ to $c \cdot f(a) + k$.

Definition For constants $h$ and $k$, the graph of

is the graph of $f$ moved $h$ units horizontally and $k$ units vertically. Each point $(a, f(a))$ goes to $(a + h, f(a) + k)$, because $x - h = a$ gives $x = a + h$. The shift is to the right when $h > 0$ and to the left when $h < 0$, and up when $k > 0$ and down when $k < 0$.

Definition The graph of $y = -f(x)$ is the reflection of the graph of $f$ across the $x$-axis, and each point $(a, b)$ goes to $(a, -b)$. The graph of $y = f(-x)$ is the reflection across the $y$-axis, and each point $(a, b)$ goes to $(-a, b)$, because $-x = a$ gives $x = -a$.

Definition For a constant $c \ne 0$, the graph of $y = c \cdot f(x)$ is obtained by multiplying every height by $c$, so each point $(a, b)$ goes to $(a, cb)$. If $\lvert c \rvert > 1$ the graph is stretched away from the $x$-axis, and if $0 < \lvert c \rvert < 1$ it is compressed toward it. If $c < 0$ the graph is also reflected across the $x$-axis. Points with $b = 0$ stay where they are, so the $x$-intercepts do not change.

Definition For a constant $b \ne 0$, the graph of $y = f(bx)$ is obtained by moving each point $(a, f(a))$ to $\left(\frac{a}{b}, f(a)\right)$, because $bx = a$ gives $x = \frac{a}{b}$. If $\lvert b \rvert > 1$ the graph is compressed toward the $y$-axis by the factor $\frac{1}{\lvert b \rvert}$, and if $0 < \lvert b \rvert < 1$ it is stretched away from it. If $b < 0$ it is also reflected across the $y$-axis. The factor must not be $0$, since $f(0 \cdot x) = f(0)$ is a constant and the graph collapses to a horizontal line.

Definition If $g(x) = c \cdot f(u(x)) + k$, then the domain of $g$ is the set of all real $x$ for which $u(x)$ lies in the domain of $f$. The whole input expression $u(x)$ must satisfy the requirement of the parent function, not just $x$. Changes outside $f$ do not change the domain, though they can change the range.

Definition For real numbers $a \ne 0$, $h$ and $k$, the graph of

is the parabola $y = x^2$ scaled vertically by $a$, then moved $h$ horizontally and $k$ vertically. Each point $(p, p^2)$ goes to $(p + h, a p^2 + k)$. The vertex $(0, 0)$ goes to $(h, k)$, the axis of symmetry is $x = h$, and the parabola opens up when $a > 0$ and down when $a < 0$. The domain stays all real numbers.

Definition If $p$ is a nonconstant polynomial and $g(x) = p(x - h) + k$, then $g$ is again a polynomial of the same degree. When $k = 0$, a number $r$ is a zero of $p$ exactly when $r + h$ is a zero of $g$. For the cubic $y = a(x - h)^3 + k$ with $a \ne 0$, the point $(0, 0)$ of $y = x^3$ goes to $(h, k)$. The domain of a polynomial is all real numbers, and shifts and scales keep it that way.

Definition For $b > 0$ with $b \ne 1$ and constants $a \ne 0$, $h$ and $k$, the graph of

has domain all real numbers and horizontal asymptote $y = k$. If $a > 0$ its range is $y > k$, and if $a < 0$ its range is $y < k$. Each point $(p, b^p)$ of $y = b^x$ goes to $(p + h, a b^p + k)$.

Definition For constants $a \ne 0$, $h$ and $k$, the graph of

is the graph of $y = \frac{1}{x}$ scaled by $a$ and moved $h$ horizontally and $k$ vertically. Its domain is $x \ne h$, because the denominator $x - h$ must not be $0$. It has vertical asymptote $x = h$, horizontal asymptote $y = k$, and range $y \ne k$.